All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
A person wearing the Ihraam by putting his upper garment below his right armpit and thus exposing his right shoulder, and covering the left shoulder, is one of the recommended acts of Tawaaf [circumambulation around the Ka'bah] according to the majority of the scholars. This way of wearing the garment is peculiar to the Tawaaf when first arriving to Makkah, and the Tawaaf of a pilgrim performing 'Umrah whilst having performed Hajj Tamatu'. Some jurists are of the view that being dressed this way is recommended for every Tawaaf after which one performs Sa'i. Therefore, according to this view, being dressed this way is also recommended for a pilgrim when performing Tawaaf on the day of 'Eed, which is called Tawaaf Al-Ifaadhah.
As regards exposing the right shoulder and covering the left one when performing Sa'i, then this is not a recommended act according to the most preponderant opinion, as no evidence has been reported on the Prophet doing so. Ibn Qudaamah reported that Imaam Ahmad said: 'We have not heard anything proving it.'
Allaah Knows best.